# Meaning of a percentage of something

I understand that a percentage is saying, how many parts there are out of 100. So 65% represents 65 parts of out 100 possible parts.

One thing that I can't get my head around, or I am over thinking, is that if we have something such as $9/12$ and I wish to know what percentage those 9 parts out of the 12 take up, what exactly am I doing? Am I saying that if the 12 parts were infact 100 parts, how much would the 9 parts now take up?

• Proportion: $\dfrac { 9 } { 12 } = \dfrac {x} {100}$. Commented Jan 23, 2018 at 13:31

We have $\frac{9}{12}=\frac{3}{4}=\frac{75}{100}$, which is evidently $75%$. This means that for every 12 objects, you get 9 objects, or put more simply, for every 4 objects you get 3 objects.

In general, given any fraction, to find its percentage, multiply the fraction by $100%$.

Here’s an easier way to approach percentages:

The sign $\%$ is similar to symbols like $e$ (Euler’s number) and $\rm m$ (the metre)—it simply stands in for another quantity—in this case, $\frac1{100}$. To solve, all you need to do is carry out the multiplication.

$$\text{P\% of X} = (P\%)(X) = P\cdot \frac1{100}\cdot X = \frac{PX}{100}$$

You probably know that, for example, $27\%=0.27$, and this is why.

To turn $\frac9{12}$ into a percentage, first divide $9$ by $12$:

$$\frac{9}{12}=0.75$$

Now, move the decimal point to the right two spaces (i.e., multiply the quotient by $100$) and then tack on the percentage sign (i.e., multiply the quotient by $\frac1{100}$):

$$\frac9{12}=0.75=75\%$$

Working from a percentage in reverse however, is not so simple. Because $$\text{percentage}={ \text{# parts} \over \text{# whole} }$$ we can never tell how many “parts” there are or what the size of the “whole” group is without more information.

For example, which country spends more on education: the United States or Costa Rica? The US spends only $4.9\%$ of its GDP, while CR spends a much greater $7.6\%$. However, the question asks you to compare actual expenditures, so percentages do not give you the entire story.

The GDP of the US is \$19.36 trillion ($ \$1.94\times10^{13}$) while the GDP of CR is only \$85.2 billion ($ \$8.52\times10^{10}$). Therefore, each country spends

$$\begin{array}{cccl} \text{US:} & (4.9\%)\left(\ 1.94\times10^{13}\right) & = & \9.506\times10^{11} \\ \text{CR:} & (7.6\%)\left(\ 8.52\times10^{10}\right) & = & \6.475\times10^{9} \\ \end{array}$$

As you can see, the percentages belie which country spends more. The moral of the story: percentages disguise the number of parts as well as the number in the whole sample. There is just no way to tell how many parts you’re dealing with without having more information.

Note, however, that sometimes we want this, because it facilitates comparisons across groups.

• I understand the calculations involving percentages, I think my issue is understand like if 3/4 is to imply 3 parts out of a possible 4, when we convert it to a percentage it becomes 75%. Is it simply just saying that if the fraction 3/4 was to be made into parts of 100, 3 parts out of 4 would equate to 75 parts out of 100? Like am i missing anything in my understand here? Commented Jan 23, 2018 at 18:25
• @salman Ah, now I understand. I’m typing up what I think will be an illuminating example. Commented Jan 23, 2018 at 19:04
• thanks, the confusion is like, 3/4ths equates to 75% but what is the underlying meaning of that? Commented Jan 23, 2018 at 19:18
• @salman See the edit I made. I think the statement in bold addresses that. Commented Jan 23, 2018 at 19:29
• so say we have 9/10 people do something. That's 90% of people doing whatever it is. What does that mean, like "90% of people" specifically. 90 out of 100 of 10 people, I'm failing to understand something or maybe I'm failing to understand what I don't understand. Commented Jan 23, 2018 at 19:51

The choice of 100 is rather arbitrary. If we had six fingers on each hand, we'd probably use pergrossages rather than percentages. Then your 9 twelfths would be 108 pergrossage.

Perhaps the most natural thing is parts of a unit, that is, parts of 1. One divided by 12 is approximately 0.083333, periodic but non-terminating in decimal. Multiply that by 9 and you should get 0.75, assuming no loss in calculating precision.

If that looks like 75%, it's only because we use the decimal system. Try this in Wolfram Alpha: 3/4 in duodecimal Then try 108 in duodecimal

$$\frac{9}{12}=\frac{3}{4}=\frac{3}{4}\frac{25}{25}=\frac{75}{100}$$

• So what does that mean, that 9 parts of the 12 relate to 75 parts of 100? Like is there anything else to it, i really feel like I am not grasping an simple yet important point. Commented Jan 23, 2018 at 13:36
• The main point is how to change the $12$ to $100$ Commented Jan 23, 2018 at 13:40

If we have something such as $\frac9{12}$ and I wish to know what percentage those $9$ parts out of the $12$ take up, what am I doing?

Don't worry. We just need to find: $$\frac9{12}\times 100 = 75\%$$ And yes, this is an another way of saying that if $12$ corresponds to a $100$, then $9$ corresponds to $75$.